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Comments on multiplicativity of maximal \(p\)-norms when \(p=2\). (English) Zbl 1162.47307

Summary: We consider the maximal \(p\)-norm associated with a completely positive map and the question of its multiplicativity under tensor products. We give a condition under which this multiplicativity holds when \(p=2\), and we describe some maps which satisfy our condition. This class includes maps for which multiplicativity is known to fail for large \(p\). Our work raises some questions of independent interest in matrix theory; these are discussed in two appendices.

MSC:

47N50 Applications of operator theory in the physical sciences
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
81R15 Operator algebra methods applied to problems in quantum theory
46N50 Applications of functional analysis in quantum physics
81P15 Quantum measurement theory, state operations, state preparations
81P68 Quantum computation
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